MinT - Best Known (85−51, 85, s)-Nets in Base 32

Best Known (85−51, 85, s)-Nets in Base 32

(85−51, 85, 142)-Net over F₃₂ — Constructive and digital

Digital (34, 85, 142)-net over F₃₂, using

1 times m-reduction [i] based on digital (34, 86, 142)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 27, 44)-net over F₃₂, using
    - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
        a function field by SÃ©mirat [i]
  2. digital (7, 59, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(85−51, 85, 273)-Net over F₃₂ — Digital

Digital (34, 85, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 85, 273)-net over F₃₂, using
- net from sequence [i] based on digital (30, 272)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273, using
    - a curve from the manYPoints database [i]

(85−51, 85, 288)-Net in Base 32 — Constructive

(34, 85, 288)-net in base 32, using

32¹ times duplication [i] based on (33, 84, 288)-net in base 32, using
- base change [i] based on digital (9, 60, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(85−51, 85, 315)-Net in Base 32

(34, 85, 315)-net in base 32, using

5 times m-reduction [i] based on (34, 90, 315)-net in base 32, using
- base change [i] based on digital (19, 75, 315)-net over F₆₄, using
  - net from sequence [i] based on digital (19, 314)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 19 and N(F) ≥ 315, using
      - a curve from the manYPoints database [i]

(85−51, 85, 37447)-Net in Base 32 — Upper bound on s

There is no (34, 85, 37448)-net in base 32, because

1 times m-reduction [i] would yield (34, 84, 37448)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 709096 361754 021018 845059 388653 036175 411743 062391 093842 785740 577103 422343 479633 192429 292567 159251 459228 931030 909875 664845 454424 > 32⁸⁴ [i]